HW1 - EML 6934 Section 6385 Fall 2009 Optimal Estimation...

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EML 6934, Section 6385 Fall 2009 Optimal Estimation University of Florida Mechanical and Aerospace Engineering HW 1 Issued: August 24, 2009, Due: August 31, 2009 (in class) Problem 1 ( Vector space). A vector space is a set that is closed under addition and scalar multi- plication 1 . That is, V is a vector space if the following are true: 1. if x and y are elements of V , then x + y V . 2. if x V , then ax V , where a is a real number. The elements of a vector space are called vectors. A familiar example of a vector space is R n , the n -dimensional real coordinate space. A subspace S of a vector space V is a subset of V that is a vector space in its own right. That is, S V and the elements of S satisfy the closure properties listed above. Consider the 2D plane (or, in fancy terminology, the vector space R 2 ). 1. Show that the set of all 2-D vectors x = [ x 1 ,x 2 ] T defined by x 1 x 2 = 0 is a subspace, but that defined by x 2 x 1 = 1 does not constitute a subspace. 2. Show that the set of real-valued continuous functions f ( x ) that are defined on the interval x [0 , 1] form a vector space. (Function addition and scalar multiplication is defined as ( f + g )( x ) = f ( x ) + g ( x ) and ( αf )( x ) = αf ( x ).) Problem 2 ( A matrix as a linear mapping). An m × n real matrix A is best thought of as a linear mapping from R n to R m . For every n -dimensional vector x that you provide to the matrix as an “input”, it produces the “output” y Δ = Ax , which is an m -dimensional vector. We write A : R n R m . Compute Ax for A = 1 2 5 4 13 6 , and express the answer in terms of the two entries x 1 ,x 2 of the input vector x . Convince yourself that the output vector y is a linear combination of the columns of A , with the input x providing the coefficients (this is true for a general matrix, not just for this example, and is a very useful fact). Problem 3 ( Range and Null spaces). The range space of an m × n matrix A , denoted by R ( A ), is the set of all vectors in R m that A can produce as output when you vary the input over all possible vectors x R n : R ( A ) = { y | Ax = y for some x } .
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